Twisted superalgebras and cohomologies of the [formula omitted] superconformal quantum mechanics

Abstract

We prove that the invariance of the N = 2 superconformal quantum mechanics is controlled by subalgebras of a given twisted superalgebra made of 6 fermionic (nilpotent) generators and 6 bosonic generators (including a central charge). The superconformal quantum mechanics actions are invariant under this quite large twisted superalgebra. On the other hand, they are fully determined by a subalgebra with only 2 fermionic and 2 bosonic (the central charge and the ghost number) generators. The invariant actions are Q i -exact ( i = 1 , 2 , … , 6 ), with a Q i ′ -exact ( i ′ ≠ i ) antecedent for all 6 fermionic generators. It follows that the superconformal quantum mechanics actions with Calogero potentials are uniquely determined even if, in its bosonic sector, the twisted superalgebra does not contain the one-dimensional conformal algebra sl ( 2 ) , but only its Borel subalgebra. The general coordinate covariance of the non-linear sigma-model for the N = 2 supersymmetric quantum mechanics in a curved target-space is fully implied only by its worldline invariance under a pair of the 6 twisted supersymmetries. The transformation connecting the ordinary and twisted formulations of the N = 2 superconformal quantum mechanics is explicitly presented.